Question

Write the sum of the order and the degree of the following differential equation-
\[\dfrac{d}{{dx}}\left( {{{\left( {\dfrac{{dy}}{{dx}}} \right)}^3}} \right) = 0\]

Answer 1

Hint: The order of the differential equation is the highest order derivative present in the differential equation and the degree is the power of the highest order derivative in the differential equation.

Complete step-by-step answer:
Here, the given differential equation is-
$ \Rightarrow \dfrac{d}{{dx}}\left( {{{\left( {\dfrac{{dy}}{{dx}}} \right)}^3}} \right) = 0$
On differentiating the given function w. r. t. x using chain rule we get,
$ \Rightarrow 3{\left( {\dfrac{{dy}}{{dx}}} \right)^{3 - 1}}\dfrac{d}{{dx}}\left( {\dfrac{{dy}}{{dx}}} \right) = 0$
As we know that $\dfrac{{d\left( {{x^n}} \right)}}{{dx}} = n{x^{n - 1}}$
On solving the above equation we get,
$ \Rightarrow 3{\left( {\dfrac{{dy}}{{dx}}} \right)^2} \times \dfrac{{{d^2}y}}{{d{x^2}}} = 0$
In this form it is easy to observe the values of degree and order of differential equations.
Here the highest order derivative is $\dfrac{{{d^2}y}}{{d{x^2}}}$ which is of second order. So the order of the differential equation is$2$.
And the degree of the highest order derivative $\dfrac{{{d^2}y}}{{d{x^2}}}$in the given differential equation is $1$ because the highest order derivative’s power is $1$ .
Here $\dfrac{{dy}}{{dx}}$ is the first order derivative hence its power is not considered.
Now we have to find the sum of order and degree of the differential equation. Then-
$ \Rightarrow $ Degree +order=$1 + 2$
$ \Rightarrow $ Sum of degree and order=$3$

Note: Differential equations are used in various disciplines from biology, economics to physics, chemistry and engineering. It is used in-
1.Model of exponential population growth of species over a long time.
2.Model of exponential decay of radioactive material.
3.Model of Newton’s law of cooling which describes the change in temperature of an object in a given environment.
4.Starting model of RL circuit to give expression of current in the circuit as a function of time.
Model for change in investment return over time.